PET467E A Note on Rock Compressibility

PET467E Rock Compressibility
A Note on Rock Compressibility
• A reservoir consists of an impervious cover or
M. Onur cap rock overlying a porous and permeable
Spring 2007 rock.

• The density differences between the oil, gas
and water phases can result in boundary
regions between them known as fluid contacts
(oil-water and gas-oil contacts).

Oil and Reservoir Pressures
Gas Solution Gas Water
• In a liquid column representing vertical pore fluid continuity the
Gas Cap
(Gas Cap) Rock pressure at any point is approximated by the relationship

Oil Gas Oil Contact pf,D=D.Gl+C
Water Oil Water Contact
(Aquifer) where D is the depth and Gl is the pressure exerted by unit height of
liquid (pressure gradient). For fresh water, Gl is 0.433 psi/ft (9.79
kpa/m, 10 m/atm). Here, C is a constant.

• Note: psi = pound per square inch

psia = absolute pressure, psig = gauge pressure

Cross-section of a combination drive type anticline
reservoir

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Rock Compressibility Depth Reservoir Pressures

• As sediment builds up, overburden pressure Pressure
increases as rock is compacted.
Overburden
• For a hydrocarbon reservoir, the reservoir
rock grains are subjected to external Fluid Grain
pressure (stress) due to overburden and to
internal pressure represented by the fluid • There is a balance in a reservoir system between the pressure
pressure.
gradients representing rock overburden pressure (pob), pore
fluids pressure (pf) and sediment grain pressure (pg):

pob = pf + pg

• pf is also called reservoir pressure and a typical value

of overden pressure (pob) is 1 psi/ft

Rock Compressibility Rock Compressibility

• As reservoir fluids are produced, the rock grains • Of principal interest to the reservoir
may expand causing a reduction in porosity which engineer is the change in the pore volume of
causes fluid to be expelled. (expansion of rock the rock or also reffered to as effective rock
grains) compressibility or formation
compressibility.
• At the same time, the overburden may cause a
decrease in the bulk volume causing fluid to be
expelled (rock compaction). In the extreme case,
we get subsidence.

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Rock Compressibility Effective Reservoir Rock
Compressibilities (Hall)
•Effective rock compressibility is
p* external pressure
=1 dV p =1 d (φVb ) p internal pressure
Vp dp φVb dp
cr =

1 (φ d (Vb ) + Vb d (φ )) = 1 d (Vb ) + 1 d (φ )
φVb dp dp Vb dp φ dp

where, p is the pore pressure, and assumes that overburden
pressure (p*) is held constant.

•Typical values are 1 to 5 times 10−6

cr = 1 ⎛ dVp ⎞ = 1 ⎛ d (φVb ) ⎞
Vp ⎜ dp ⎟ φVb ⎜ dp ⎟
⎝ ⎠ p* ⎝ ⎠ p*

Formation Compaction Component of Rock Compressibility
Total Rock Compressibility (Hall)
• Three kinds of compressibility can be
distinguished in rocks: (a) rock matrix
compressibility (cs), (b) rock bulk volume
compressibility (cb), (c) pore compressibility (cr)

cb = (1−φ)cs + φcr

p* external pressure cc = −1 ⎛ dVp ⎞ = − 1 ⎛ d (φVb ) ⎞
p internal pressure Vp ⎜ dp* ⎟ φVb ⎜ dp* ⎟
⎝ ⎠p ⎝ ⎠p

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Rock Compressibility A Derivation For Rock/Pore/Bulk
Compressibilities
• Rock matrix compressibility (cs) is the fractional
change in volume of the solid rock material with a • For a fixed external pressure, and varying internal fluid
unit change in internal pressure
pressure (as is done in the experiments conducted by Hall,
• Rock bulk compressibility (cb) is the fractional we may derive the following expression:
change of bulk volume of the rock with a unit
change in internal pressure. cb = φcr + (1−φ ) cs

• Pore compressibility (cr) is the fractional change cr = 1 ⎛1 ∂Vb ⎞ (1−φ ) ⎛ 1 ∂Vs ⎞ = 1 ∂Vp
of pore volume of rock with a unit change in φ ⎜ ∂p ⎟− ⎜ ∂p ⎟ Vp ∂p
internal pressure. ⎝ Vb ⎠ φ ⎝ Vs ⎠

cr = cb − (1−φ ) cs
φ
φ

A Derivation For Rock/Pore/Bulk A Derivation For Rock/Pore/Bulk
Compressibilities Compressibilities

• According to Geerstma (1957), we should treat variations of • Geerstma derives:

bulk volume and pore volue (or porosity) with respect to both 1 ⎛ ∂Vb ⎞ = 1 ⎛ ∂Vb ⎞ − cm
external pressure, p* and internal pressure, p. Vb ⎜ ∂p ⎟ Vb ⎜ ∂p* ⎟
⎝ ⎠ p* ⎝ ⎠p

dVp ( p*, p) = ⎛ ∂Vp ⎞ dp* + ⎛ ∂Vp ⎞ dp 1 ⎛ ∂Vp ⎞ = 1 ⎡ 1 ⎛ ∂Vb ⎞ ⎤
⎜ ∂p* ⎟ ⎜ ∂p ⎟ Vp ⎜ ∂p* ⎟ φ ⎢ ⎜ ∂p* ⎟ − cm ⎥
⎝ ⎠p ⎝ ⎠ p* ⎝ ⎠p ⎣⎢Vb ⎝ ⎠p
⎥⎦

dVb ( p*, p) = ⎛ ∂Vb ⎞ dp* + ⎛ ∂Vb ⎞ dp 1 ⎛ ∂φ ⎞ −1 ⎛ ∂φ ⎞ (1−φ ) ⎛ ∂Vb ⎞ cm
⎜ ∂p* ⎟ ⎜ ∂p ⎟ φ ⎜ ∂p* ⎟ φ ⎜ ∂p ⎟ ⎜ ∂p* ⎟ φ
⎝ ⎠p ⎝ ⎠ p* ⎝ ⎠p = ⎝ ⎠ p* = Vbφ ⎝ ⎠p −

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A Derivation For Rock/Pore/Bulk Notes
Compressibilities
• We can predict change of porosity with pressure
• Geerstma definitions: from:

= 1 ⎛ ∂Vb ⎞ ; = 1 ⎛ ∂Vp ⎞ φ( p2 ) = φ( p1)[1+ cr ( p2 − p1 )]
cb Vb ⎜ ∂p* ⎟ cr Vp ⎜ ∂p* ⎟
⎝ ⎠p ⎝ ⎠p or simply

cm =1 dVm =1 dVm = cs φ2 = φ1[1+ cr ( p2 − p1 )]
Vm dp Vm dp*
• As fluid pressure decreases to a value p2 from an
= 1 ⎛ ∂φ ⎞ = −1 ⎛ ∂φ ⎞ = (1−φ ) − cm initial pressure p1, porosity decreases slightly.
cp φ ⎜ ∂p* ⎟ φ ⎜ ∂p ⎟ cb φ
⎝ ⎠p ⎝ ⎠ p* φ

= 1 [cb − cm ]
cr φ

Note on Reservoir Temperatures Note on Reservoir Temperatures

• Reservoir temperature may be expected to • A 5000-ft well in a region with an average

confirm to the regional or local geothermal ambient (surface) temperature of 65oF
gradient. In many basins, this is around may be expected to have a bottom-hole
0.034 K/m (2 oF/100 ft). temperature of roughly

• A figure of 4 oF/100 ft is a high gradient, 65 + 5000 x (2/100) = 165oF

whereas 0.5 oF/100 ft is low. • It is a reasonable assumption that

reservoir condition processes tend to be
isothermal (temperature is constant
throughout the reservoir).

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Correlations for cr

• It is usually the one of the difficult
parameters to obtain and effective rock
compressibilities are rarely measured.

• There are several correlations for cr
depending on the rock type (see Horne’s

book “Modern Well Test Analysis).

( )cr = exp 4.026 − 23.07φ + 44.28φ 2 x10−6 Consolidated limestones
( )cr = exp 5.118 − 36.26φ + 63.98φ 2 x10−6 Consolidated sandstones

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